Question

Number of ordered pairs of positive integers having 30 as their l.C.M is

Answer 1

we have to find number of ordered pair of positive integers having 30 as their LCM

assume , N = 30

prime factors of N = 2¹ × 3¹ × 5¹

we know, if $N=a^{k_1}\times b^{k_2}\times c^{k_3}$
then, number of sets of (x , y) = $\frac{(2k_1+1)(2k_2+1)(2k_3+1)-1}{2}+1$

so, number of sets of (x,y) = {(2 × 1 + 1)(2 × 1 + 1)(2 × 1 + 1) - 1}/2 + 1
= {27 - 1}/2 + 1
= 14

so, number of ordered pairs = 2 × 14 - 1 = 27